Export armature animation + corrective smooth modifier?
Answer is quite simple: you cannot. Corrective smoothness is very advanced effect that simply cannot be exported, nor baked at the moment.

Only way to bake that I can think of, anyway, would be (with e.g. a py script), for each frame, to make a diff between the mesh with and without CS modifier, and store that diff in a relative shapekey, and then animate all those skeys such that they each express at one frame.

But this sounds like an awfully heavy solution to me. And result is not much guaranteed even, since shapekeys are applied before the armature modifier (and the whole modifier stack), while Corrective smoothness modifier comes after the armature modifier…